C*-envelopes for operator algebras with a coaction and co-universal C*-algebras for product systems

Research output: Contribution to journalJournal articlepeer-review

  • A. Dor-On
  • E. T.A. Kakariadis
  • E. Katsoulis
  • M. Laca
  • X. Li

A cosystem consists of a possibly nonselfadoint operator algebra equipped with a coaction by a discrete group. We introduce the concept of C*-envelope for a cosystem; roughly speaking, this is the smallest C*-algebraic cosystem that contains an equivariant completely isometric copy of the original one. We show that the C*-envelope for a cosystem always exists and we explain how it relates to the usual C*-envelope. We then show that for compactly aligned product systems over group-embeddable right LCM semigroups, the C*-envelope is co-universal, in the sense of Carlsen, Larsen, Sims and Vittadello, for the Fock tensor algebra equipped with its natural coaction. This yields the existence of a co-universal C*-algebra, generalizing previous results of Carlsen, Larsen, Sims and Vittadello, and of Dor-On and Katsoulis. We also realize the C*-envelope of the tensor algebra as the reduced cross sectional algebra of a Fell bundle introduced by Sehnem, which, under a mild assumption of normality, we then identify with the quotient of the Fock algebra by the image of Sehnem's strong covariance ideal. In another application, we obtain a reduced Hao-Ng isomorphism theorem for the co-universal algebras.

Original languageEnglish
Article number108286
JournalAdvances in Mathematics
Number of pages40
Publication statusPublished - 2022

Bibliographical note

Publisher Copyright:
© 2022 Elsevier Inc.

    Research areas

  • C*-envelope, Co-universal algebra, Coaction, Covariance algebra, Nica-Pimsner algebras, Product systems


ID: 310973613